An adaptive moving grid method for a system of singularly perturbed initial value problems

• We consider an adaptive moving grid approach to solve a system of first-order singularly perturbed initial value.
• A priori error analysis in maximum norm is constructed.
• The mesh is constructed adaptively by equidistributing a monitor function.
• A first-order rate of convergence, independent of all perturbation parameters is established.

A system of first-order singularly perturbed initial value problems is considered. The system is discretized by a backward Euler difference scheme for which a priori error analysis in the maximum norm is constructed. It is shown from the a priori error bound that there exists a mesh with NN subintervals that gives optimal error bound of O(N−1)O(N−1) which is robust with respect to the perturbation parameters. A partly heuristic argument based on a priori error analysis leads to a suitable monitor function. Based on an a posteriori error bound, a first-order rate of convergence, independent of all perturbation parameters, is established. A linear and a nonlinear examples are tested, and the numerical results are provided to demonstrate the effectiveness of our adaptive moving grid method.